NOC:Geometry of Curves and Surfaces

Lecture 1 - Parametrized Curves and Differentiable Curves

Lecture 2 - Length of a Curve

Lecture 3 - Regular Curve, Arc-length, Re-parametrization of a Curve

Lecture 4 - Unit Speed Re-parametrization, Curvature of a Curve

Lecture 5 - Curvature of a regular curve and Curvature formulas

Lecture 6 - Level curves, Implicit function theorem

Lecture 7 - Orientation of a curve, Signed curvature, Tangent and Normal vectors

Lecture 8 - Turning Angle

Lecture 9 - Curve determined by curvature, Total Curvature

Lecture 10 - Hopf's Rotation Index Theorem

Lecture 11 - Four Vertex Theorem

Lecture 12 - Isoperimetric Inequality

Lecture 13 - Space Curves and Principal Vectors

Lecture 14 - Frenet-Serret Equations

Lecture 15 - Curve determined by curvature and torsion

Lecture 16 - Several variable calculus

Lecture 17 - Regular surfaces in real three dimensional space

Lecture 18 - Examples of regular surfaces

Lecture 19 - Regular Value, Critical Value of a differentiable map

Lecture 20 - Regular surface locally graph of a differentiable function

Lecture 21 - Change of parametrizations in regular surfaces

Lecture 22 - Differentiable function on surfaces

Lecture 23 - Surface of Revolution, Torus, Higher genus surfaces

Lecture 24 - Tangent Plane and Tangent Vectors

Lecture 25 - Examples of Tangent Planes

Lecture 26 - Differential Map between Tangent Spaces

Lecture 27 - Orientation of Surfaces and Normal Vector Field

Lecture 28 - First Fundamental Form on Surfaces

Lecture 29 - Change of Parametrization in First Fundamental Form and Area

Lecture 30 - Area of Some Surfaces and Gradient on Surfaces

Lecture 31 - Isometries, local isometries, conformal maps

Lecture 32 - Gauss Map and its Fundamental Properties

Lecture 33 - Self Adjoint Operator

Lecture 34 - Second Fundamental Form and normal Curvature

Lecture 35 - Examples for second fundamental form

Lecture 36 - Principal curvatures and Principal Directions

Lecture 37 - Examples for principal curvatures and principal directions

Lecture 38 - Gaussian Curvature and Mean Curvature

Lecture 39 - Christoffel Symbols and The Gauss's Remarkable Theorem

Lecture 40 - Invariance of Gaussian Curvature and Examples of Christoffel Symbols

Lecture 41 - Vector Field and Covariant Derivative

Lecture 42 - Vector Field along a curve, parallel vector field and geodesics

Lecture 43 - Geodesics and its equations

Lecture 44 - Simplicial Complex, Triangulation and Euler Characteristics

Lecture 45 - Gauss Bonnet Theorem